Equivalent norms,continous functions etc..

Hi,
I was going through the book "Matrix Analysis and Applied Linear Algebra" by Carl D Meyer and was having a good time.
Suddenly I came across vector norms and equivalent norms, and my brain shut down.
So, I did some searching and found what looks like a nice article explaining this madness:Proof That All Norms On Finite Vector Space Are Equivalent

It first shows that if two norms are equivalent on the unit sphere then they are equivalent everywhere. I understand that to some extent.

Then it goes and shows that if we are working with the 2-norm, any other norm is a continous function with respect to the 2-norm.
I have a question regarding this.
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Take an arbitrary finite-dimensional space and an arbitrary norm .
Also suppose that is a basis of and so an element of may be written as .
Now given an , choose such that implies that

How does all that imply that the maximum difference between two components of vectors x and y is less than the ratio of a number larger than 0 ( ), and the sum of the norms of the basis vectors ?